central university of rajasthan semester-wise scheme...

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ANNEXURE: X (Pages: 01-24)

CENTRAL UNIVERSITY OF RAJASTHAN

Semester-Wise Scheme and Syllabus of

M.Sc. B.Ed. Mathematics (3 Year) 2018-2019 to Onward Semester – I

S.

No.

Course

Code

Course Title Type of

Course (C/E)

L T P Credits

1 MBM411 Abstract Algebra C 3 1 0 4

2 MBM412 Real Analysis C 3 1 0 4

3 MBM413 Mathematical Programming C 3 1 0 4

4 MBM414 Qualitative Theory of ODE C 3 1 0 4

5 ED101 Basics of Education C 3 0 0 3

6 ED102 Senior Secondary Education in India:

Status, Challenges & Strategies

C 3 0 0 3

Total 18 4 0 22

Semester – II

S.

No.

Course

Code


Course (C/E)

L T P Credits

1 MBM421 Linear Algebra C 3 1 0 4

2 MBM422 Complex Analysis C 3 1 0 4

3 MBM423 Topology C 3 1 0 4

4 MBM424 PDE C 3 1 0 4

5 ED201 Philosophy of Mathematics C 3 0 0 3

6 ED202 Learner and Learning C 3 0 0 3

Total 18 4 0 22

Semester – III

S.

No.

Course

Code


Course (C/E)

L T P Credits

1 MBM531 Functional Analysis C 3 1 0 4

2 MBM532 Numerical Analysis C 3 1 0 4

3 MBM533 Measure Theory & Integration C 3 1 0 4

4 MBM534 Elective Paper E 3 1 0 4

5 ED301 Teaching Approaches and Strategies C 3 0 0 3

6 ED302 Pedagogy of Mathematics C 4 0 0 4

Total 19 4 0 23

Semester – IV

S.

No.

Course

Code


Course (C/E)

L T P Credits

1 MBM541 Mathematical Modeling C 3 1 0 4

2 MBM542 Dynamics of Rigid Body C 3 1 0 4

3 MBM543 Elective Paper E 3 1 0 4

4 ED401 Learning Assessment C 3 0 0 3

5 ED402 Pedagogical Analysis of Senior

Secondary School: Mathematics

C 4 0 0 4

6 ED403 Classroom Organization & Management C 3 0 0 3

Total 19 3 0 22

2

Semester – V

S.

No.

Course

Code


Course (C/E)

L T P Credits

1 ED501 Internship-I C 0 0 8 8

2 ED502 Internship-II C 0 0 12 12

Total 0 0 20 20

Semester – V

S.

No.

Course

Code


Course (C/E)

L T P Credits

1 MBM561 Project/Dissertation in

Mathematics/Elective/Open Elective

E/OE 0 0 16 16

2 ED601 Project/Dissertation in Education C 0 0 4 4

Total 0 0 20 20

Total Grand- 109 Credits

Note- MSM: M.Sc. in Mathematics; Numeric (xyz) in paper/course code: x-Level of Course, y- Semester, z- Paper/Course Number; C-

Compulsory/Core Course; E- Elective, O/OE- Open Elective; 15L- 15 Lectures; LTP: Lecture-Tutorial-Practical

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List of Elective/Open Electives Papers of M.Sc. B.Ed. Mathematics (3 Year)

(Semester Wise: 2018-2019 to Onward)

Sr.

No.

Elective Course Title Credits Semester &

Course Code

Remarks

1 Mathematical Software Tools 4 Semester-III

(MBM534)

Out of these two elective papers, any one may

be taken for course code MBM534 2 Programming in C 4

3 Celestial Mechanics 4 Semester-IV

(MBM543)

Out of these eight elective papers, any one

may be taken for course code MBM543 4 Computational ODE 4

5 Game Theory 4

6 Graph Theory 4

7 Integral Equations and Calculus of Variations 4

8 Number Theory-I 4

9 Operations Research 4

10 Special Functions 4

11 Advance Complex Analysis 4

Semester-VI

(MBM561)

Interested students may opt open elective

paper for code MBM561 12 Advance Real Analysis 4

13 Advanced Numerical Methods 4

14 Automata Theory and Formal Languages 4

15 Bio-Mathematics 4

16 Celestial Mechanics 4

17 Complex Dynamics 4

18 Computational PDE 4

19 Dynamical Systems 4

20 Financial Mathematics 4

21 Fluid Dynamics 4

21 Fractional Calculus and Geometric Function Theory 4

23 Fuzzy Logics and Its Applications 4

24 Module Theory 4

25 Nonlinear Dynamics and its application to

Information Technology

4

26 Number Theory-II 4

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Detail Revised Syllabus for M. Sc. B.Ed. Mathematics (3 Year)

(2018-2019 to Onward)

SEMESTER-I

1. MBM411: ABSTRACT ALGEBRA LTL: 3+1+0

Unit I: Review of groups, subgroups, normal subgroups, quotient groups, homomorphism, isomorphism theorems,

conjugacy relation, Group Action, Stabilizer and orbits, Class equation, Sylow theorems, Normal and subnormal

series, composition series, Jordan holder theorem. Solvable groups, Simple groups, simplicity of An. (15 L)

Unit II: Rings, homomorphisms, ideals, Quotient rings, prime ideals, maximal ideals, field of quotients of an

integral domain, Euclidean rings, unique factorization domains, principal ideal domain. Prime and Irreducible

elements, GCD, Polynomial rings. Chinese remainder theorem for rings and PID‟s, Eisenstenin‟s criterion of

irreducibility. (15 L)

Unit III: Extension fields, algebraic element and transcendental elements, Simple extension, Primitive elements,

Algebraic extension, Roots of a polynomial and splitting of a polynomial over fields. (15 L)

Recommended Reading:

1. Topics in algebra by I. N. Herstein. Wiley Eastern Limited.

2. A first course in Abstract Algebra by John Fraleigh (3rd Edition), Narossa Publishing House.

3. Basic Abstract Algebra by Bhattacharya, Jain and Nagpal, 2nd Edition.

4. Algebra Vol 1 & 2 by Ramji Lal, Springer 2017

5. Algebra by S.Mclane and G.Birkhoff, 2nd Edition,

6. Basic Algebra by N.Jacbson, Hind.Pub.Corp.1984.

2. MBM412: REAL ANALYSIS LTP: 3+1+0

Unit-I: Euclidian space ℝn: Open ball and open sets, closed sets, adherent points, accumulation points, closure of

sets, derived sets, Bolzano Weierstrass theorem. Cantor intersection theorem. Lindeloff covering theorem, Heine

Borel theorem, Compactness in ℝn. Metric spaces: open sets, closed sets, compact subsets of a metric space.

(15L)

Unit-II: Functions of bounded variations: Monotonic functions and its properties, types of discontinuity functions

of bounded variations and its properties, total variations. Functions of several variables: continuity, partial

derivatives, differentiability, derivatives of functions in an open set of ℝn into ℝn

as a linear transformations, chain

rule, Taylor‟s theorem, inverse function theorem, implicit function theorem and explicit function theorem,

Jacobians. (15L)

Unit-III: Riemann-Stieltjes integral: Definition and existence of R-S integration, conditions of R-S integrability,

properties of R-S integrals, integration and differentiation. Sequence and series of functions: Pointwise and

uniform convergence, Uniform convergence and continuity, Uniform convergence and integration, Uniform

convergence and differentiation, Uniform convergence and R-S integration. (15L)

5


1. W. Rudin , Principles of Mathematical Analysis (3rd

Ed.)McGraw Hill International Edition, 1976

2. Mathematical Analysis by T. M. Apostol( 2nd

Ed.), Narosa Publishing House , 1985

3. Theory of Functions of a Real Variable, Volume 1 by I. P. Natanson, Frederick Pub. Co., 1964

4. H.L. Royden, Real Analysis, McMillan Publication Co. Inc. New York

5. Malik, S.C. Mathematical Analysis, Wiley Eastern, New Delhi, 1984.

3. MBM413: MATHEMATICAL PROGRAMMING LTP: 3+1+0

UNIT-I: Linear Programming, Theoretical foundation of Simplex Method, Revised Simplex Method, Duality in

linear programming problem, Primal-dual solution relationship, , Duality theorems. Dual simplex method.; Post

optimality analysis. (15 L)

UNITI-I: Integer Linear programming: Gomory's Cutting Plane Method, Branch & Bound Method. Multi-

objective optimization theory. Goal Programming. Dynamic Programming, application of dynamic programming,

Bellman‟s principle of optimality. (15 L)

UNIT-III: Nonlinear programming, Solution of nonlinear programming problem with equality constraints and with

not all equality constraints, Kuhn-Tucker necessary and sufficient conditions for optimality of the objective

function in NLPP. Quadratic programming, Wolfe‟s method and Beale‟s Method (15 L)

Recommended Reading:.

1. Kanti Swarup, P. K. Gupta and Manmohan: Operations Research, S. Chand & Co.

2. Hamady Taha: Operations Research, Mac Millan Co.

3. S. D. Sharma: Nonlinear and Dynamic Programming, Kedar Nath Ram Nath & Co.

4. G. Hadley: Linear Programming, Oxford and IBH Publishing Co.

5. S. I. Gass: Linear Programming, Mc Graw Hill Book Co.

4. MBM414: QUALITATIVE THEORY OF ORDINARY DIFFERENTIAL EQUATIONS LTP:3+1+0

UNIT-I: Existence and uniqueness of solutions of initial value problems for first order ordinary differential equations:

separation and comparison theorems, system of equations existence theorems, Homogeneous linear systems, Non

homogeneous linear systems, and linear systems with constant coefficients, Singular solutions of first order ODEs. (15 L)

UNIT-II: Cauchy-Euler equation, Two-point boundary-value problem, variation of parameters, Green's functions,

Construction of Green's functions, Non homogeneous boundary conditions, Orthogonal sets of function, Strum-Liouville

boundary value problem, Eigen values and Eigen functions, Eigen function expansions convergence in the mean. (15L)

UNIT-III: Autonomous system of differential equations, Critical points and Stability for Linear systems with constant

coefficients, linear plane autonomous systems, perturbed systems, Method of Lyapunov for nonlinear systems. Limit cycles of

Poincare-Bendixson Theorem. (15 L)

Recommended Reading: 1. Simmons: Ordinary Differential Equations.

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2. Lakshmikantham, Deo and Raghavendra, Ordinary Differential Equations.

3. Simmons, G.F. and Krantz, S.G., Differential Equations: Theory, Technique, and Practics, TMH Publishing.

5. ED101: BASICS OF EDUCATION LTP: 3+0+0

6. ED101: SENIOR SECONDARY EDUCATION IN INDIA: STATUS, CHALLENGES AND STRATEGIES LTP: 3+0+0

SEMESTER-II

7. MBM421: LINEAR ALGEBRA LTP: 3+1+0

Unit I: Review of Vector Spaces, The algebra of linear transformations, Isomorphism, Linear functional, dual and double

dual, Eigen values and Eigen vectors, Annihilating polynomials, diagonalization, Tringularization. (15 L)

Unit II: Determinants and its geometric properties, Laplace expansion, Rational and Jordan canonical form, primary

decomposition theorem, nilpotent matrices, canonical form for nilpotent matrix, computation of invariant factors, non-

negatives matrices, generalized inverse of a matrix (15 L)

Unit III: Symmetric and Skew-symmetric Bilinear Forms, Diagonalization of symmetric bilinear forms. Inner Product Space,

The adjoint of Linear Transformation, Unitary operators, Self adjoints and Normal Operators. (15 L)


1. Algebra by S. Mclane and G. Birkhoff

2. Linear algebra by S. Lang, Springer

3. Linear Algebra by Bisht and Sahai

4. Linear Algebra by Hoffman and Kunze, P.H.I

5. Matrix Analysis and Applied linear Algebra, by Carl D. Meyer

6. Linear Algebra by S. Lang. 7. Linear Algebra by P. Lax.

7. Linear Algebra: a geometric approach by S. Kumaresan

8. MBM422: COMPLEX ANALYSIS LTP: 3+1+0

UNIT-I: Functions of a complex Variable, Differentiability and analyticity, Cauchy Riemann Equations, Power

series as an analytic function, properties of line integrals, Goursat Theorem, Cauchy theorem, consequence of

simply connectivity, index of a closed curve (15 L)

UNIT -II: Cauchy‟s integral formula, Morera‟s theorem, Liouville‟s theorem, Fundamental theorem of Algebra,

Harmonic functions, Existence of Harmonic conjugate, Taylor‟s theorem, Zeros of Analytic functions,

Laurent series, singularities, classification of singularities (15 L)

UNIT -III: Maximum modulus theorem, Minimum modulus theorem, Hadamard three circle theorem, Schwarz‟s

Lemma, Rouche‟s theorem, Calculation of residues, Residue theorem, Evaluation of integrals of the form

∫

, ∫

, Conformal mappings. (15 L)


1. Complex Analysis ( Third edition) by L. V. Ahlfors, McGraw Hill Book Company, 1979

7

2. Complex Analysis by J. B. Conway, Narosa Publishing House,

3. Complex Analysis by Serg Lang, Addison Wesley

4. Foundations of Complex analysis (Second Edition), S. Ponnusamy, Narosa Publishing House.

5. Complex variables and Applications by Ruel V. Churchill.

9. MBM423: TOPOLOGY LTP: 3+1+0

UNIT -I: Topological spaces. Open sets, closed sets. Interior points, Closure points. Limit points, Boundary points,

exterior points of a set, Closure of a set, Derived set, Dense subsets. Basis, sub base, relative topology. (15 L)

UNIT -II: Continuous functions, open & closed functions, homeomorphism, Lindelof„s, Separable spaces,

Connected Spaces, locally connectedness, Connectedness on the real line, Components, Compact Spaces, one point

compactification, compact sets, properties of Compactness and Connectedness under a continuous functions,

Compactness and finite intersection property, Equivalence of Compactness. (15 L)

UNIT -III: Separation Axioms: T0 , T1, and T2 spaces, examples and basic properties, First and Second Countable

Spaces, Regular, normal, T3 & T4 spaces, Tychnoff spaces, Urysohn‟s Lemma, Tietze Extension Theorem, finite

product topological spaces and some properties. (15 L)


1. G.F.Simmons: Topology and Modern Analysis, McGraw Hill (1963)

2. W. J. Pervin, Foundations of General Topology

3. Willard, Topology, Academic press

4. Vicker , Topology via logic (School of Computing, Imperial College, London)

5. Topology, A First Course By: J. R. Munkers Prentice Hall of India Pvt. Ltd.

10. MBM424: PARTIAL DIFFERENTIAL EQUATION LTP: 3+1+0

UNIT-I: Formation of PDEs: First order PDE in two and more independent variables, Derivation of PDE by elimination

method of arbitrary constants and arbitrary functions. Lagrange‟s first order linear PDEs, Charpit‟s method for non-linear

PDE of first order, Monge‟s method, Jacobi‟s method and Cauchy problem for first order PDEs. (15L)

UNIT-II: PDEs of second order with variable coefficients: Classification of second order PDEs, Canonical form, Parabolic,

Elliptic and Hyperbolic PDEs, Method of separation of variables for Laplace, Heat and Wave equations, Eigen values and

Eigen functions of BVP, Orthogonality of Eigen function, , D‟ Almbert‟s solutions to wave equations. (15L)

UNIT-III: General solution of higher order PDEs, Fundamental solution of Laplace Equation, Green‟s function for Laplace

Equation, Wave equation, Diffusion Equation, Solution of BVP in spherical and cylindrical coordinates, Variational

formulation of boundary value problem. (15L)

Recommended Reading: 1. K, Sankara, Rao, Introduction to Partial Differential Equations, Phi Learning. 2. Ian N. Sneddon, Elements of Partial Differential Equations, Dover Publications. 3. Garrett Birkhoff and Gian-Carlo Rota, Ordinary Differential Equations.

11. ED201: PHILOSOPHY OF MATHEMATICS LTP: 3+0+0

12. ED202: LEARNER AND LEARNING LTP: 3+0+0

8

SEMESTER-III

13. MBM531: FUNCTIONAL ANALYSIS LTP: 3+1+0

UNIT-I: Inner product spaces, Normed linear spaces, Banach spaces, Quotient norm spaces, continuous linear

transformations, equivalent norms, the Hahn-Banach theorem and its consequences. Conjugate space and

separability, second conjugate space, Weak *topology on the conjugate space (15 L)

UNIT-II: The natural embedding of the normed linear space in its second conjugate space, The open mapping

Theorem, The closed graph theorem, The conjugate of an operator, The uniform boundedness principle, Definition

and examples of a Hilbert space and simple properties, orthogonal sets and complements (15 L)

UNIT-III: The projection theorem, separable Hilbert spaces. Bessel's inequality, the conjugate space, Riesz's

theorem, The adjoint of an operator, self adjoint operators, Normal and unitary operators, Projections, Eigen values

and eigenvectors of on operator on a Hilbert space, Spectral theorem on a finite dimensional Hilbert space. (15 L)

Recommended Reading: 1. G.F.Simmons: Topology and Modern Analysis, McGraw Hill (1963)

2. G.Bachman and Narici : Functional Analysis, Academic Press 1964

3. A.E.Taylor : Introduction to Functional analysis, John Wiley and sons (1958)

4. A.L.Brown and Page : Elements of Functional Analysis, Van-Nastrand Reinehold Com

5.B.V. Limaye: Functional Analysis, New age international.

6.Erwin Kreyszig, Introductory functional analysis with application, Willey.

14. MBM532. NUMERICAL ANALYSIS LTP: 3+1+0

UNIT-I: Polynomial equations-Descartes' rule of signs, Sturm sequence, root of a polynomial equation-iterative

and direct methods. System of linear equations: error analysis for direct methods-Gauss elimination and Gauss-

Jordan elimination method, convergence analysis for iterative methods- Gauss-Jacobi iterative method and Gauss-

Seidel iterative method(15 L)

UNIT-II: Hermite interpolation, piecewise and spline interpolation, cubic spline interpolation, numerical

differentiation-methods based on interpolation and methods based on finite difference operators, methods based on

undetermined coefficients, errors in numerical differentiation, numerical integration-methods based on interpolation

and methods based on undetermined coefficients, composite integration methods, double integration.(15 L)

UNIT-III: Numerical solution of ordinary differential equations: initial value problems, existence and uniqueness

of the solution of initial value problem, single step method-Taylor series, Picard's method, Euler's method, modified

Euler method, Runge-Kutta method, multi-step method-Adam-Bashforth method, Adams-Moulton methods,

stability analysis, predictor-corrector method, eigenvalue problem, power method, Jacobi method, Householder

method.(15 L)


1. K.E. Atkinson, “An Introduction to Numerical Analysis”, 2nd

Ed.,Wiley-India, 1989.

2. J. I. Buchaman and P. R. Turner, “Numerical Methods and Analysis”, McGraw-Hill, 1992.

3. M.K. Jain, S.R.K. Iyengar and R.K. Jain, “Numerical Methods for Scientific and Engineering

Computation”, 6th

Ed., New Age International Publishers, 2012.

4. S. S. Sastry, “Introduction Methods of Numerical Analysis”, 4th

Ed., Prentice-Hall of India, 2006.

9

15. MBM533: MEASURE THEORY AND INTEGRATIONS LTP:3+1+0

UNIT-I: Countable and uncountable sets, cardinality and cardinal arithmetic, Schr ̈der–Bernstein theorem,

the Canter‟s ternary set, semi-algebras, algebras, monotone class, algebras, measure and

outer measures, Carathe ̈dory extension process of extending a measure on a semi-algebra to generated

algebras, Borel sets (15 L)

Unit-II: Lebesgue outer measure and Lebesgue measure on R, translation invariance of Lebesgue measure,

existence of a non-measurable set, characterizations of Lebesgue measurable sets, the Cantor-Lebesgue function,

measurable functions on a measure space and their properties, Borel and Lebesgue measurable functions, Simple

functions and their integrals, Littlewood‟s three principle (statement only) (15 L)

UNIT-III: Lebesgue integral on R and its properties, bounded convergence theorem, Fatou‟s lemma, Lebesgue

monotone convergence theorem, Lebesgue dominated convergence theorem, L_p-spaces, Holder-Minkowski

inequalities, parseval‟s identity, Riesz Fisher‟s theorem. (15 L)


1. H. L. Royden amd P. M. Fitzpatrick, Real Analysis (Fourth edition), PHI 2010.

2. P. R. Halmos, Measure Theory, Springer, 1994.

3. E. Hewit and K. Stromberg, Real and Abstract Analysis, Springer, 1975.

4. K. R. Parthasarathy, Introduction to Probability and Measure, Hindustan Book Agency, 2005.

5. I. K. Rana, An Introduction to Measure and Integration (2nd Edition) Narosa Publishing House, 2005.

16. MBM534: ELECTIVE LTP: 3+1+0

17. ED301: TEACHING APPROACHES AND STRATEGIES LTP: 3+0+0

18. ED301: PEDAGOGY OF MATHEMATICS LTP: 3+1+0

SEMESTER-IV

19. MBM541: MATHEMATICAL MODELING LTP: 3+1+0

UNIT-I: Introduction to modeling. Definition of System, classification of systems, classification and limitations of

mathematical models, Methodology of model building, Modeling through ordinary differential equation: linear growth and

decay models, non-linear growth and decay models, Compartment models. (15 L)

UNIT-II: Checking model validity, verification of models, Stability analysis, Basic model relevant to population dynamics,

Epidemics modeling, Ecology, Environment Biology through ordinary differential equation, Partial diff.equation. (15 L)

UNIT-III: Basic theory of linear difference equations with constant coefficients, Mathematical modeling through difference

equations in population dynamics, genetics, Markov chains model, Gambler‟s ruin model, Stochastic models, Monte Carlo

methods . (15 L)

Recommended Reading: 1. D. N. P. Murthy, N. W. Page and E. Y. Rodin, Mathematical Modeling, Pergamon Press.

2. J. N. Kapoor, Mathematical Modeling, Wiley Estern Ltd.

3. P. Fishwick: Simulation Model Design and Execution, PHI, 1995, ISBN 0-13-098609-7

4. A. M. Law, W. D. Kelton: Simulation Modeling and Analysis, McGraw-Hill, 1991, ISBN 0-07-100803-9

5. J. A. Payne, Introduction to Simulation, Programming Techniques and Methods of Analysis, Tata McGraw Hill Publishing

Co. Ltd.

10

6. F. Charlton, Ordinary Differential and Differential equation, Van Nostarnd.

20. MBM542: DYNAMICS OF RIGID BODY LTP: 3+1+0

UNIT-I: Moments and products of inertia, moment of inertia of a body about a line through the origin, Momental

ellipsoid, rotation of co-ordinate axes, principal axes and principal moments. K.E. of rigid body rotating about a

fixed points, angular momentum of a rigid body, Eulerian angle, angular velocity, K.E. and angular momentum in

terms of Eulerian angle. Euler‟s equations of motion for a rigid body, rotating about a fixed point, torque free

motion of a symmetrical rigid body (rotational motion of Earth). (15L)

UNIT-II: Classification of dynamical systems, Generalized co-ordinates systems, geometrical equations,

Lagrange‟s equation for a simple system using D‟Alembert principle, Deduction of equation of energy, deduction

of Euler‟s dynamical equations from Lagrange‟s equations, Hamilton‟s equations, Ignorable co-ordinates,

Routhian Function. (15L)

UNIT-III: Hamiltonian‟s principle for a conservative system, principle of least action, Hamilton-Jacobi equation,

Phase space and Liouville‟s Theorem, Canonical transformation and its properties, Lagrange Brackets, and passion

brackets, Poisson-Jacobi identity. (15L)


1. Vectorial Mechanics by E. A. Milne; Methuen & Co. Ltd. London, 1965.

2. Dynamics (Part II) by A.S. Ramsey; CBS Publishers & Distributors, Delhi, 1985.

3. A treatise on Analytical Dynamics by L.A. Pars; Heinemann, London, 1968.

4. Classical mechanics by H. Goldstein; arosa Publishing House, New Delhi, 1990.

5. Generalized Motion of Rigid Body by N. Kumar; Narosa Publishing House, New Delhi, 2004.

6. Classical Mechanics by K. Sankara Rao; PHI learning Private Ltd., New Delhi, 2009

21. MBM543: ELECTIVE LTP: 3+1+0

22. ED401: LEARNING ASSESSMENT LTP: 3+0+0

23. ED402: PEDAGOGICAL ANALYSIS OF SENIOR SECONDARY SCHOOL: MATHEMATICS LTP: 3+1+0

24. ED403: CLASSROOM ORGANIZATION AND MANAGEMENT LTP:3+1+0

SEMESTER-V

25. ED501: INTERNSHIP-I LTP: 0+0+8

26. ED502: INTERNSHIP-II LTP: 0+0+12

SEMESTER-VI

27. MBM561: PROJECT DISSERTAION IN MATHEMATICS LTP:0+0+16

28. ED601: PROJECT DISSERTAION IN EDUCATION LTP:0+0+4

11

DETAIL OF REVISED ELECTIVE/OPEN ELECTIVE COURSES IN M.Sc. B.Ed. MATHEMATICS (3 YEAR)

(2018-2019 to Onward)

Sr. No. Elective Course Title Credits

1. Advance Complex Analysis 4

2. Advance Real Analysis 4

3. Advanced Numerical Methods 4

4 Automata Theory and Formal Languages 4

5 Bio-Mathematics 4

6 Celestial Mechanics 4

7 Complex Dynamics 4

8 Computational ODE 4

9 Computational PDE 4

10 Differential Geometry 4

11 Dynamical Systems 4

12 Fluid Dynamics 4

13 Financial Mathematics 4

14 Fractional Calculus and Geometric Function Theory 4

15 Fuzzy Logics and Its Applications 4

16 Game Theory 4

17 Graph Theory 4

18 Integral Equations and Calculus of Variations 4

19 Mathematical Tools and Software 4

20 Module Theory 4

21 Nonlinear Dynamics and its application to Information Technology 4

22 Number Theory-I 4

23 Number Theory-II 4

24 Operations Research 4

25 Programming in C 4

26 Special Functions 4

12

Detail Revised Syllabus of Elective Papers

1. ADVANCED COMPLEX ANALYSIS LTP: 3+1+0

UNIT-I: Analytic Continuation, Analytic Continuation along Paths via Power Series, Monodromy Theorem, Picard theorem,

Poisson integral, Mean value theorem, Schwarz reflection principle, Analytic continuation via reflexion. (15 L)

UNIT-II: Infinite sums and infinite product of complex numbers, Infinite product of analytic functions, Factorization of entire

functions, The Gamma functions, The Zeta functions. (15 L)

UNIT-III: The Riemann mapping theorem (Statement only), Area Theorem, Biberbach Theorem and conjecture, Distortion

theorem, Koebe ¼ theorem, Starlike and convex functions. Coefficient estimates and distortion theorem. (15 L)


1. S. Ponnusamy, Foundation of Complex Analysis, 2nd edition, Narosa Publishing House.

2. L. R. Ahlofrs, Complex Analysis, McGraw Hill

2. ADVANCED REAL ANALYSIS LTP: 3+1+0

UNIT-I: Metric spaces revisited; Baire Category theorem, completion of Metric spaces, Banach contraction

principle and some of its applications. Compactness, Total boundedness, characterization of compactness for

arbitrary Metric spaces; Arzella-Ascoli theorem, Stone Weierstrass theorem. (15L)

UNIT-II: Integrations : Lebesgue‟s criterion of Riemann integrability over a bounded closed interval [a, b] and its

consequence, length of a rectifiable curve in a plane, Riemann-Stieltjes integral over [a, b] and its properties,

Integrators of bounded variation, Integration by parts, Stieltjes integral as a Riemann integral, Step function as

integrator, Riesz theorem. (15L)

UNIT-III: Cesaro‟s Method of Summability and Fourier Series: Cesaro‟s method of summability of order 1 and

order 2, Some specific examples, Regularity of Cesaro‟s method, Definition of Fourier series and some examples,

Dirichlet‟s Kernel, Fejer‟s Kernel, Fejer‟s theorem, Dini‟s and Jordan‟s tests for point wise convergence of Fourier

series. (15L)


[1] A. M. Bruckner, J. Bruckner & B. Thomson : Real Analysis, Prentice-Hall, N.Y. 1997.

[2] R. R. Goldberg : Methods of Real Analysis, Oxford-IBH, New Delhi, 1970.

[3] I. P. Natanson : Theory of Functions of a Real Variable, Vol-I, F.Ungar, N.Y. 1955.

[4] E. Hewitt and K. Stromberg : Real and Abstract Analysis, John-Willey, N.Y. 1965.

[5] J. F. Randolph : Basic Real and Abstract Analysis. Academic Press, N.Y. 1968.

[6] P. K. Jain and K. Ahmad : Metric Spaces, Narosa Publishing House.

[7] G. Tolstov : Fourier Series, Dover Publication, N.Y. 1962.

13

3. ADVANCED NUMERICAL METHODS LTP: 3+1+0

UNIT-I: Numerical solution of algebraic and transcendental equations: Introduction- iteration method, Newton-

Raphson method, Graeffe‟s root square method, acceleration of convergence. Numerical Solution of systems of

nonlinear equations: iteration method, Newton-Raphson method. Linear Systems of equations: Introduction- Gauss

elimination method, LU decomposition, Solution of tridiagonal system, Ill-conditioned linear systems and method

for Ill-conditioned matrix. Eigen Value problem: Power method, Jacobi Method, Householder method. (15L)

UNIT-II: Polynomial Interpolation: introduction- finite difference formulas, divided difference interpolation,

Aitken‟s formula, Hermite‟s interpolation, double interpolation, Spline interpolation (linear, quadratic and cubic

spline), Error in cubic Spline. Numerical differentiation, Errors in numerical differentiation, cubic spline method;

Numerical Integration: introduction to trapezoidal, Simpson‟s rules and error estimates, use of cubic splines,

numerical double integration. (15L)

UNIT-III: Boundary value problem: Introduction, BVP governed by second order ordinary differential equations,

Finite difference method, shooting method, cubic splines method. IVP and BVP in partial differential equations:

classification of linear second order partial differential equations, Finite difference methods for Laplace and

Poisson equations - Jacobi method, Gauss-Seidel method and ADI (alternating direction implicit) method , Finite

difference method for heat conduction equation - Bender- Schmidt recurrence relation, Crank-Nicolson formula,

and Jacobi Iteration formula, Finite difference method for wave equation. (15L)


1. K. E. Atkinson: An Introduction to Numerical Analysis.

2. J. I. Buchaman and P. R. Turner: Numerical Methods and Analysis.

3. S. S. Sastry: Introductory Methods of Numerical Analysis.

4. S. R. K. Iyengar and P. K. Jain: Numerical Methods.

4. AUTOMATA THEORY AND FORMAL LANGUAGES LTP:3+1+0

Unit-I: Theory of Computation: Finite automata, Deterministic and non-deterministic finite automata, equivalence

of deterministic and non-deterministic automata, Moore and Mealy machines, Regular expressions, Grammars and

Languages, Derivations, Language generated by a grammar. (15L)

Unit-II: Regular Language and regular grammar, Regular and Context free grammar, Context sensitive grammars

and Languages, Pumping Lemma, Kleene‟s theorem. (15L)

Unit-III: Turing Machines: Basic definitions, Turing machines as language acceptors, Universal Turing machines,

decidability, undecidability, Turing Machine halting problem. (15L)


1) D. Kelly, Automata and Formal Languages: An Introduction, Prentice-Hall, 1995.

2) J. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata, Languages, and Computation

14

(2nd

edition), Pearson Edition, 2001.

3) P. Linz, An Introduction to Formal Languages and Automata, 3rd Edition.

5. BIO- MATHEMATICS LTP: 3+1+0

Unit 1: Introduction: Goals and Challenges of mathematical modeling in biology and ecology. Idealization and

general principle of model building, deterministic and stochastic models, different types of mathematical models

and differential and difference equations as relevant mathematical techniques, complex network dynamics,

biological and ecological examples. (15 L)

Unit 2: Continuous growth models for single species: The linear model, Logistic population model, Stability of

equilibrium states and bifurcation analysis, Constant Harvesting and bifurcations, Delay models, Linear analysis of

delay models: periodic solutions. (15 L)

Unit 3: Discrete population models for single species: Simple models, Discrete logistic-type model: Chaos,

stability, periodic solutions and bifurcations, discrete delay models. Models with interacting populations: predator-

prey interactions, Analysis of a predator-prey model with limit cycle periodic behaviour, Harvesting in two species

models. (15 L)


1. J.D. Murray, Mathematical biology: An introduction, Springer, 2007.

2. F. Brauer, C.C-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2000.

3. N.F. Britton, Essential mathematical biology, Springer, 2004.

4. M. Kot, Elements of mathematical ecology, Cambridge University Press, 2001.

5. A. Okubo, S.A. Levin, Diffusion and ecological problems, Springer, 2002.

6. S.V. Petrovskii, B.L. Li, Exactly solvable models of biological invasions, CRC Press/ Chapman and Hall,

2005.

7. R.W. Sterner, J.J. Elser, Ecological stoichiometry: the biology of elements from molecules to the biosphere,

Princeton University Press, 2002.

8. H. Smith, An Introduction to Delay Differential Equations with Applications to Life Sciences, Springer,

2010

6. CELESTIAL MECHANICS LTP: 3+1+0

UNIT-I: Orbital Motion Introduction, Kepler‟s Laws of Planetary Motion, Newton‟s law of gravitation, Central

force motion, Integral of energy, Differential equation of orbit, Inverse square force, Geometry of orbits, Two body problem, Motion of center of mass, Relative motion, Solution of two-body problem, Earth bound satellite circular

orbit, Classical orbital elements, Kepler‟s equation, Position in elliptic orbit, Position in parabolic orbit and Position in hyperbolic orbit. (15 L)

UNIT-II: Three-body Problem and N-body problem Motion of the center of mass, The angular momentum, or

areal velocity, integrals, Integrals of energy, Zero Velocity Curves or Zero Velocity Surfaces, Equation of relative motion, Stationary solution of three body problem, Restricted three body problem- formulation and its solution,

Stability of motion near Lagrangian points. (15 L)

15

UNIT-III: Rocket Dynamics Introduction to Rocket Dynamics, Equation of motion for variable mass, Performance of a Single-Stage Rocket, Exhaust speed parameters, Effect of gravity, performance of a Two-Stage

Rocket, Optimization of a Multi-Stage Rocket. (15 L)


1. Introduction to Celestial Mechanics by S. W. McCuskey, Addison-Wesley Publishing Company, 1963.

2. Solar System Dynamics by C. D. Murray and S. F. Dermott, Cambridge University Press, 2000.

3. An Introduction to Celestial Mechanics by F. R. Moulton, the MacMillan Company, 1914.

4. Theory of orbits. The Restricted problem of three bodies by V. Szebehely, New York Acad. Press, 1967.

5. Classical Mechanics by K. Sankara Rao, PHI Learning Pvt. Ltd., 2009

7. COMPLEX DYNAMICS LTP: 3+1+0

UNIT –I: Iteration of a Mobius transformation, attracting, repelling and indifferent fixed points. Iterations of R(z)

= z2, z

2+c, z +

. The extended complex plane, chordal metric, spherical metric, rational maps, Lipschitz condition,

conjugacy classes of rational maps, valency of a function, fixed points, Critical points, Riemann Hurwitz relation.

(15 L) UNIT –II: Equicontinuous functions, normality sets , Fatou sets and Julia sets, completely invariant sets, Normal

families and equicontinuity, Properties of Julia sets, exceptional points Backward orbit, minimal property of Julia

sets. (15 L)

UNIT -III: Julia sets of commuting rational functions, structure of Fatou set, Topology of the Sphere, Completely

invariant components of the Fatou set , The Euler characteristic, Riemann Hurwitz formula for covering maps,

maps between components of the Fatou sets, the number of components of Fatou sets, components of Julia sets.

(15L)


1. A. F. Beardon, Iteration of rational functions, Springer Verlag , New York, 1991.

2. L. Carleson and T . W. Gamelin, Complex dynamics, Springer Verlag, 1993.

3. S. Morosawa, Y. Nishimura, M. Taniguchi, T. Ueda, Holomorphic dynamics,

Cambridge University Press, 2000.

4. X. H. Hua, C. C. Yang, Dynamics of transcendental functions, Gordan and Breach Science

Pub. 1998.

.

8. COMPUTATIONAL ODE LTP: 3+1+0 UNIT-I: Numerical solutions of system of simultaneous first order differential equations and second order initial value

problems (IVP) by Euler and Runge-Kutta (IV order) explicit methods, Numerical solutions of second order boundary value

problems (BVP) of first, second and third types by shooting method. (15L)

UNIT-II: Types Finite difference schemes of second order BVP based on difference operators (solutions of tri-diagonal

system of equations), Solutions of such BVP by Newton-Cotes and Gaussian integration rules, Convergence and stability of

finite difference schemes. (15L)

UNIT-III: Variational principle, approximate solutions of second order BVP of first kind by Reyleigh-Ritz,

Galerkin, Collocation and finite difference methods, Finite Element methods for BVP-line segment, triangular and

rectangular elements, Ritz and Galerkin approximation over an element, assembly of element equations and

imposition of boundary conditions. (15L)

Recommended Reading: 1. M. K. Jain, S. R. K. Iyenger and R. K. Jain, Numerical Methods for Scientific and Engineering Computations,

New Age Publications, 2003.

16

2. M. K. Jain, Numerical Solution of Differential Equations, 2nd

edition, Wiley-Eastern.

3. S. S. Sastry, Introductory Methods of Numerical Analysis,

4. D.V. Griffiths and I. M. Smith, Numerical Methods for Engineers, Oxford University Press, 1993.

5. C. F. Gerald and P. O. Wheatley, Applied Numerical Analysis, Addison- Wesley, 1998.

6. A. S. Gupta, Text Book on Calculas of Variation, Prentice-Hall of India, 2002.

7. Naveen Kumar, An Elementary Course on Variational Problems in Calculus, Narosa, 2004.

9. COMPUTATIONAL PDE LTP: 3+1+0

UNIT-I: Numerical solutions of parabolic equations of second order in one space variable with constant coefficients:-

two and three levels explicit and implicit difference schemes, truncation errors and stability, Difference schemes for

diffusion convection equation, Numerical solution of parabolic equations of second order in two space variable with

constant coefficients-improved explicit schemes, Implicit methods, alternating direction implicit (ADI) methods. (15L)

UNIT-II: Numerical solution of hyperbolic equations of second order in one and two space variables with constant and

variable coefficients-explicit and implicit methods, alternating direction implicit (ADI) methods. (15L)

UNIT-III: Numerical solutions of elliptic equations, Solutions of Dirichlet, Neumann and mixed type problems with

Laplace and Poisson equations in rectangular, circular and triangular regions, Finite element methods for Laplace,

Poisson, heat flow and wave equations.

Recommended Reading: 1. M. K. Jain, S. R. K. Iyenger and R. K. Jain, Computational Methods for Partial Differential Equations, Wiley

Eastern, 1994.

2. M. K. Jain, Numerical Solution of Differential Equations, 2nd

edition, Wiley Eastern.

3. S. S. Sastry, Introductory Methods of Numerical Analysis, , Prentice-Hall of India, 2002.

4. D. V. Griffiths and I. M. Smith, Numerical Methods of Engineers, Oxford University Press, 1993.

5. C. F. General and P. O. Wheatley, Applied Numerical Analysis, Addison- Wesley, 1998.

6. K. J. Bathe and E. L. Wilson, Numerical Methods in Finite Element Analysis, Prentice-HalI, 1987.

7. A. S. Gupta, Text Book on Calculas of Variation, Prentice-Hall of India, 2002.

8. Naveen Kumar, An Elementary Course on Variational Problems in Calculus, Narosa, 2004.

10. DIFFERENTIAL GEOMETRY LTP: 3+1+0

UNIT-I : Differential Calculus, Tangent space. Vector fields, Cotangent space and differentials on Charts and

atlases. Differential manifolds, Induced topology on manifolds, functions and maps, some special functions of

class Para compact manifolds and partition of unity. Pullback functions, local coordinates systems and

partial derivatives. (15 L)

UNIT-II : Tangent vectors and Tangent space, differential of a map, the tangent bundle, pullback vector fields, Lie

bracket, the cotangent space, the cotangent bundle, the dual of the differential map. One parameter group and

vector fields. (15 L)

17

UNIT-III : Lie derivatives, tensors, tensor fields, connections, parallel translation, covariant differentiation of

tensor fields, torsion tensor, curvature tensor, Bianchi and Ricci identities , Geodesics, Riemannian manifolds.(15

L)


1. K. S. Amur, D. J. Shetty, C. S. Bagewadi, An introduction to differential geometry, Narosa Publishing house,

2010.

2. B. O‟Neill, Elementary differential geometry, Academic Press , New York, 1966

3. J. A. Thorpe, Elementary topics in differential geometry, Undergraduate text in Mathematics, Springer Verlag ,

1979.

4. T. J. Willmore, An introduction to differential geometry, Oxford University Press, 1965.

5. U. C. De, A. A. Shaikh, Differential geometry of Manifolds, Narosa Pub. House, 2009

6. D. Somasundaram, Differential Geometry: a first course, Narosa Pub. House, 2010.

11. DYNAMICAL SYSTEMS LTP: 3+1+0

Unit-I Linear Systems: Exponentials of operators, Linear systems in R2, Complex eigenvalues, Multiple

eigenvalues, Jordon forms, Stability theory, generalized eigenvectors and invariant subspaces, Non-homogeneous

linear systems. (15L)

Unit-II: Non-linear Systems: local analysis: the fundamental existence-uniqueness theorem, The flow defined by a

differential equation, Linearization, The stable manifold theorem, The Hartman-Grobman theorem, Stability and

Liapunov functions, Saddles, Nodes, Foci, and Centers. (15L)

Unit-III: Non-linear Systems: global analysis: Dynamical systems and global existence theorem, Limit sets and

Attractors, Periodic orbits, Limit Cycles, and Seperatrix cycles, the Poincare map, the stable manifold theorem for

periodic orbits, the Poincare-Bendixon theory in R2, Lineard Systems, Bendixon‟s Criteria. (15L)


1. Differential Equations and Dynamical Systems by Lawrence Perko, Springer-Verlag, 2006.

2. Differential Equations, Dynamical Systems and an Introduction to Chaos by Morris W. Hirsch, Stephen

Smale and Robert L. Devaney, Academic Press, 2013

3. Dynamical Systems and Numerical Analysis by A.M. Stuart and A.R. Humphries, Cambridge University

Press, 1998.

4. Dynamical Systems with Applications using MATLAB by S. Lynch, Birkhause press, 2004.

5. Nonlinear Dynamics and Chaos with Applications to Physics, Biology, Chemistry and Engineering, by

Steven H. Strogatz, Westview Press.

12. FLUID DYNAMICS LTP: 3+1+0

UNIT-I: Physical Properties of fluids. Concept of fluids, Continuum Hypothesis, density, specific weight, specific

volume, Kinematics of Fluids : Eulerian and Lagrangian methods of description of fluids, Equivalence of Eulerian

and Lagrangian method, General motion of fluid element, integrability and compatibility conditions, strain rate

tensor, stream line, path line, streak lines, stream function¸ vortex lines, circulation. (15L)

UNIT-II: Stresses in Fluids : Stress tensor, symmetry of stress tensor, transformation of stress components from

one co-ordinate system to another, principle axes and principle values of stress tensor Conservation Laws :

Equation of conservation of mass, equation of conservation of momentum, Navier Stokes equation, equation of

18

moments of momentum, Equation of energy, Basic equations in different co-ordinate systems, boundary conditions. (15L)

UNIT-III: Irrotational and Rotational Flows : Bernoulli‟s equation, Bernoulli‟s equation for irrotational flows, Two

dimensional irrotational incompressible flows, Blasius theorm, Circle theorem, sources and sinks, sources sinks and

doublets in two dimensional flows, methods of images. (15L)


1. An introduction to fluid dynamics, R.K. Rathy, Oxford and IBH Publishing Co. 1976.

2. Theoretical Hydrodynamics, L. N. Milne Thomson, Macmillan and Co. Ltd.

3. Textbook of fluid dynamics, F. Chorlton, CBS Publishers, Delhi.

4. Fluid Mechanics, L. D. Landau and E.N. Lipschitz, Pergamon Press, London, 1985.

13. FINANCIAL MATHEMATICS LTP: 3+1+0

UNIT-I: Introduction to options and markets: types of options, interest rates and present values, Black Sholes

model : arbitrage, option values, pay offs and strategies, putcall parity, Black Scholes equation, similarity solution

and exact formulae for European options, American option, call and put options, free boundary problem. (15L)

UNIT-II: Binomial methods: option valuation, dividend paying stock, general formulation and implementation,

Monte Carlo simulation : valuation by simulation, Lab component: implementation of the option pricing algorithms

and evaluations for Indian companies. (15L)

UNIT-III: Finite difference methods: explicit and implicit methods with stability and conversions analysis methods

for American options- constrained matrix problem, projected SOR, time stepping algorithms with convergence and

numerical examples. (15L)


1. D. G. Luenberger, Investment Science, Oxford University Press, 1998. 2. J. C. Hull , Options, Futures and Other Derivatives, 4th ed., Prentice- Hall ,New York, 2000.

3. J. C. Cox and M. Rubinstein, Option Market, Englewood Cliffs, N. J.: Prentice-Hall, 1985.

4. C.P. Jones. Investments, Analysis and Measurement, 5th ed., John Wiley and Sons, 1996.

14. FRACTIONAL CALCULUS AND GEOMETRIC FUNCTION THEORY LTP: 3+1+0

UNIT-I: Fractional derivatives and Integrals, application of fractional calculus, Laplace transforms of fractional

integrals and fractional derivatives, fractional ordinary differential equations, fractional integral equations, Initial

value problem of fractional differential equations. (15 L)

UNIT-II: Univalence in Complex plane. Area theorem. Growth, covering and distortion results. Starlike and

Convex functions. Starlike and Convex functions of order α. Alpha convexity. Close to convexity, spirallikeness

and Φ-likeness in unit disk. (15 L)

UNIT-III: Subordination. Application of subordination principle. First and second order differential subordination.

Briot-Bouquet differential subordinations. Briot-Bouquet application in Univalent function theory. (15 L)

Recommended Reading: 1. Loknath Debnath and Dambaru Bhatta, Intgral Transforms and Special Functions, CRC press, 2010.

2. I. Graham and G. Kohr, Geometric function theory in one and higher dimensions, Marcel Dekker, 2003.

3. S. S. Miller and P. T. Mocanu, Differential Subordinations theory and Applications, Marcel Dekker, 2000.

15. FUZZY LOGIC AND ITS APPLICATIONS LTP:3+1+0

19

UNIT–I: Classical sets vs Fuzzy Sets- Definition and Mathematical representations, Level Sets, Operations on fuzzy sets,

Compliment, Intersections, Unions, Combinations of Operations, Fuzzy negation, triangular norms, t-conorms, Aggregation

Operations, max-min principle, extension principle, Fuzzy Arithmetic, Fuzzy Relations, Fuzzy Binary and n-ary relations,

composition of fuzzy relations. (15L)

UNIT–II: Fuzzy Numbers, Linguistic Variables, Arithmetic Operations on intervals & Numbers, Classical Logic,

Multivalued Logics, Fuzzy Propositions, Fuzzy Qualifiers, Linguistic Hedges Fuzzy Measures, Evidence Theory, Necessity

and Belief Measures, Probability Measures vs Possibility Measures. (15L)

UNIT-III: Fuzzy knowledge base, rule base, Data base for fuzzy, Inference rules, Fuzzification, defuzzification, Fuzzy

Decision Making-Individual, Multi Person, Multistage, Multi criterion Decision Making, Fuzzy Linear Programming, Fuzzy

Classification and Pattern Recognition. (15L)


1. G. J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice Hall, 1997.

2. Timothy J.Ross, “Fuzzy logic with engineering applications”, McGraw Hill, 1995.

3. H. –J. Zimmermann, Fuzzy Set Theory and Its Application, 2nd ed., Kluwer, Boston, 1990. K.H.Lee.. First

Course on Fuzzy Theory and Applications, Springer-Verlag.

4. J. Yen and R. Langari.. Fuzzy Logic, Intelligence, Control and Information, Pearson Education.

16. GAME THEORY LTP: 3+1+0

UNIT-I: A General Introduction to Game Theory-its Origin, Representation of Games, Types of Game, Static

Games with Complete and Incomplete Information, Strategic Form Game with Illustrations, Solution Concept- Pure

and Mixed Strategies, Dominance and Best Response, Pareto Optimality, Maxmin and Minmax Strategies, Pure and

Mixed Strategies Nash Equilibrium, Correlated Equilibrium, Bayesian Games, Market Equilibrium and Pricing:

Cournot and Bertrand Game. (15L)

UNIT-II: Existence and Properties of Nash Equilibrium, Two-person Zero-Sum Games-its Solution; Dynamic

Games of Perfect Information, Extensive Form Game, Nash Equilibrium, Sub-game Perfection, Backward

Induction (looking forward), Stackelberg Model of Duopoly. (15L)

UNIT-III; Bargaining Problem, Dynamic Games with Imperfect Information, Finitely and Infinitely Repeated

Games, The Folk Theorem, Illustrations, Stochastic Games, Coalition Games, Core and Shapley Value,

Illustrations. (15L)


1. M.J. Osborne, An Introduction in Game Theory, Indian Ed.

2. M. J. Osborne and A. Rubinstein, A course in Game Theory, MIT Press, 1994

3. D. Fudenberg and J. Tirole, Game Theory, MIT Press, 1991.

4. J. Von Neumann and O. Morgenstern, Theory of Games and Economic Behaviour, New York: John Wiley

and Sons., 1944.

5. R.D. Luce and H. Raiffa, Games and Decisions, New York: John Wiley and Sons.,1957.

6. G. Owen, Game Theory, (Second Edition), New York: Academic Press, 1982.

17. GRAPH THEORY LTP: 3+1+0

20

UNIT-I: Graphs and simple graphs, Vertex Degrees, Subgraphs, Graph Isomorphism, The Incidence and

Adjacency Matrices, Paths and Circuits, Trees, Cut Edges and Cut Vertices, Euler and Hemilton circuits, Bipartite

and Complete graphs. Spanning trees, Minimal spanning trees, Kruskal‟s Algorithm, Directed graphs, Weighted

undirected graphs, Dijkstra‟s algorithm, Warshal‟s Algorithm. (15L)

UNIT-II: Connectivity: Connectivity of graphs, Cut-sets, Edge Connectivity and Vertex Connectivity, Planarity:

Planar Graphs, Testing of Planarity, Euler‟s formula for connected planar graphs, Kuratowski Theorem for Planar

graphs, Random Graphs. (15L)

UNIT-III: Coloring of graphs: Chromatic number and chromatic polynomial of graphs, Brook‟s Theorem, Five

Color Theorem and Four Color Theorem. (15L)

Recommended Reading: 1) F. Harary, Graph Theory, Narosa Publ.

2) R. Diestel, Graph Theory, Springer, 2000.

3) Narsingh Deo, Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall of India.

4) W. T. Tutte, Graph Theory, Cambridge University Press, 2001.

5) Kenneth H. Rosen, Discrete Mathematics and its Applications, 6th

ed., Tata McGraw-Hill.

18. INTEGRAL EQUATIONS AND CALCULUS OF VARIATIONS LTP: 3+1+0

UNIT-I: The variation of a functional and its properties, Euler‟s equations and application, Geodesics, Variational

problems for functional involving several dependent variables, Hamilton principle, Variational problems with

moving (or free) boundaries, Approximate solution of boundary value problem by Rayleigh-Ritz method. (15 L)

UNIT-II: Linear integral equation and classification of conditions, Volterra integral equation, Relationship between

linear differential equation and Volterra integral equation, Resolvent kernel of Volterra integral equation, solution

of integral equation by Resolvent kernel, The method of successive approxi., Convolution type equation. (15 L)

UNIT-III: Fredholm integral equation, Fredholm equation of the second kind, Fundamentals-iterated kernels,

constructing the resolvent kernel with the aid of iterated kernels, Integral equation with degenerated kernels,

solutions of homogeneous integral equation with degenerated kernel. (15 L)


1. Applied Mathematics for Engineers and Physicists by L. A. Pipe (McGraw Hill)

2. Introduction to Mathematical Physics by Charlie Harper, P.H.I. , New Delhi

3. Higher Engineering Mathematics by B.S. Grewal, Khanna Publications, Delhi

4. Mathematical Methods for Physicists by George Arfken (Academic Press)

5. Mathematical Methods by Potter and Goldberg (Prentice Hall of India)

6. Calculus of Variations by IM Gelfand, SV Fomin, and Richard A Silverman

7. Introduction to the Calculus of Variations by Bernard Dacorogna, World Scientific

8. Calculus of Variations with Applications to Physics and Engineering by Robert Weinstock, Dover Publications.

9. Linear Integral equations By R. P. Kanwal, Academic Press, New- York.

19. MATHEMATICAL TOOLS AND SOFTWARE LTP: 3+0+1

UNIT-I: MATLAB: Basic Introduction: Simple arithmetic calculations, Creating and working with arrays,

numbers and matrices, Creating and printing simple plots, Function files, Applications to Ordinary differential

equations: A first order ODE, A 2nd order ODE, ode23, ode45, Basic 2-D plots and 3-D plots. (10 L)

21

UNIT-II: Mathematica: Basic introduction: Arithmetic operations, functions, Graphics: 2-D plots, 3-D plots,

Plotting the graphs of different functions, Matrix operations, Finding roots of an equation, Finding roots of a system

of equations, Solving differential equations. (10 L)

UNIT-III: LaTeX: Basic Introduction: Mathematical symbols and commands, Arrays, Formulas, and Equations,

Spacing, Borders and Colors, Using date and time option in LaTeX, To create applications and Letters, PPT in

LaTeX, Writing an article, Pictures and Graphics in LaTeX. (10L)

(Note- Lab for 1 credits= 30 hrs.)


1. R. Pratap: Getting started with MATLAB, Oxford University Press, 2010.

2. S. Lynch, Dynamical Systems with Applications using MATLAB, Birkhäuser, 2014.

3. M. L. Abell, J.P. Braselton, Differential Equations with Mathematica, Elsevier Academic Press, 2004.

4. I. P. Stavroulakis, S.A. Tersian, An Introduction with Mathematica and MAPLE, World Scientific, 2004.

5. L.W. Lamport, LaTeX: A document Preparation Systems, Addison-Wesley Publishing Company, 1994.

6. H. Kopka, P.W. Daly, Guide to LATEX, Fourth Edition, Addison Wesley, 2004.

7.

20. MODULE THEORY LTP: 3+1+0

UNIT-I: Modules over a ring, submodules, Quotient Modules, module homomorphism and isomorphism theorems

for modules, cyclic modules, simple modules and semisimple modules and rings, Schur‟s lemma. (15L)

UNIT-II: Exact sequences, Products, Coproducts and their universal property, External and internal direct sums,

Free modules, Left exactness of Hom sequences and counter-examples for non-right exactness. (15L)

UNIT-III: Noetherian and Artinian modules and rings. Hilbert basis theorem, Projective and injective modules,

Divisible groups, Example of injective modules. (15L)

Recommended Reading: 1. I. N. Herstein, Topics in Algebra, Wiley Eastern, 1975.

2. P. B. Bhattacharya, S. K. Jain and S. R. Nagpaul, Basic Abstract Algebra (2nd Edition), Cambridge

University Press, 1997.

3. D. S. Malik, J. N. Mordeson, and M. K. Sen, Fundamentals of Abstract Algebra, McGraw-Hill

International Edition, 1997.

4. D.S. Dummit and R.M. Foote, Abstract Algebra, John Wiley, 2003.

5. J.S. Golan, Modules & the Structures of Rings, Marcl Dekkar. Inc.

21. NONLINEAR DYNAMICS AND ITS APPS. TO INFORMATION TECHNOLOGY LTP: 3+1+0

UNIT-I: Introduction to nonlinear Dynamics. Non linear Maps. Bass Model for technology Diffusion (in both

Differential and Difference form). (15 L)

UNIT-II: Existence of chaos in Logistic and Base Models in Discrete forms. Effects of variable externals and

internal influence in Bass Model (in both continuous and discrete forms, conditions for chaos). (15 L)

22

UNIT-III: Modelling of two or more competing Technologies and their coexistence, Application to markets.

Models for Virus in Communication and Computer Networks. Network Crimes and Control. (15 L)

Recommended Reading: 1. P. Glendenning, Stability, Instability and Chaos, Cambridge University Press (1994).

2. M. Laxshmanan and S. Rajsekher, Nonlinear Dynamics, Springer-Verlag, Heidelberg (2003)

22. NUMBER THEORY-I LTP: 3+1+0

UNIT-I: Primes, Divisibility, Greatest common divisor, Euclidean algorithm, Fundamental theorem of arithmetic,

Perfect numbers, Mersenne primes and Fermat numbers, Farey sequences. (15L)

UNIT-II: Congruence and modular arithmetic, Residue classes and reduced residue classes, Chinese remainder

theorem, Fermat's little theorem, Wilson's theorem, Euler's theorem and its application to cryptography, Arithmetic

functions , Möbius inversion formula, Greatest integer function. (15L)

UNIT-III: Primitive roots and indices, quadratic residues, Legendre symbol, Euler's criterion, Gauss's lemma,

Quadratic reciprocity law, Jacobi symbol, Representation of an integer as a sum of two and four squares,

Diophantine equations ax+by=c, x2+y

2=z

2, x

4+y

4=z

4. Binary quadratic forms and Equivalence of quadratic forms.

(15L)


1.David M. Burton, Elementary Number Theory, Wm. C. Brown Publishers, Dubuque, Iowa 1989.

2. G.A. Jones and J.M. Jones, Elementary Number Theory, Springer-Verlag, 1998.

3. W. Sierpinski, Elementary Theory of Numbers, North-Holland, Ireland, 1988.

4. Niven, S.H. Zuckerman and L.H. Montgomery, An Introduction to the Theory of

Numbers, John Wiley, 1991.

5. Joseph H. Silverman, A Friendly Introduction to Number Theory, 4th

ed., Pearson.

6. Thomas Koshy, Elementary Number Theory with Applications, 2nd

ed., Academic Press.

23. NUMBER THEORY – II LTP: 3+1+0

UNIT-I: Continued fractions, Approximation of real numbers by rational numbers, Pell's equations, Partitions,

Ferrers graphs, Jacobi's triple product identity. (15L)

UNIT-II: Congruence properties of p(n), Rogers-Ramanujan identities, Minkowski's theorem in geometry of

numbers and its applications to Diophantine inequalities, Order of magnitude and average order of arithmetic

functions. (15L)

UNIT-III: Euler's summation formula, Abel's identity, Elementary results on distribution of primes, Characters of

finite Abelian groups, Dirichlet's theorem on primes in arithmetical progression. (15L)


1. Thomas Koshy, Elementary Number Theory with Applications, 2nd

ed., Academic Press.

2. Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976.

23

24. OPERATIONS RESEARCH LTP: 3+1+0

UNIT-I: Nonlinear Programming: Unconstrained algorithms; direct search method, gradient method . Constrained

methods; Separable programming, quadratic programming. General Inventory models, role of demand in the

development of inventory; Static Economic-Order-Quantity (EOQ) models; Dynamic EOQ models. Continuous

review model, single period models, multi period models. (15 L)

UNIT-II: Elements of Queuing models, role of exponential, pure birth and death models. Generalized Poisson

Queuing models, Specialized Poisson Queues: Steady state measures of performance, single server model multi

server models, machine servicing models-(M/M/R): (GD/K/K), R< K.

Replacement and maintenance models; gradual failure, sudden failure, replacement due to efficiency deteriorate

with time, staffing problems, equipment renewal problems. (15 L)

UNIT-III: Project Scheduling by PERT-CPM Simulation modelling: Monte Carle Simulation, Types of

simulations, Elements of discrete-events simulation, generation of random numbers. Mechanics of discrete

simulation, Methods of gathering statistical observations: subinterval method, replication method, regeneration

method. Sequencing Problems: notions, terminology, and assumptions, processing n jobs through m machines. (15

L)


1. Operations Research an Introduction –Hamady A. Taha, Prentice Hall.

2. Operations Research Theory and Applications-J.K.Sharma, Macmillan Publishers.

3. Non linear Programming –S.D. Sharma, Kedar Nath Ram Nath & Co.

4. Mathematical Programming Theory and Methods-S.M.Sinha.

5. Operations Research, - Kanti Swarup, P. K. Gupta and Man Mohan, Sultan Chand & Sons

25. PROGRAMMING IN C LTP: 3+0+1

UNIT-I: Basic concepts of programming languages: Programming domains, language evaluation criterion and

language categories, Describing Syntax and Semantics, formal methods of describing syntax, recursive descent

parsing, attribute grammars, Dynamic semantics (operational semantics, denotational semantics, axiomatic

semantics) (10 L)

UNIT-II: Names, Variables, Binding, Type checking, Scope and lifetime data types, array types, record types,

union types, set types and pointer types, arithmetic expressions, type conversions, relational and Boolean

expressions, assignment statements, mixed mode assignment. Statement level control structures, compound

statements, selection statement, iterative statements, unconditional branching. (10 L)

UNIT-III: Programming in C: Character set, variables and constants, keywords, Instructions, assignment

statements, arithmetic expression, comment statements, simple input and output, Boolean expressions, Relational

operators, logical operators, control structures, decision control structure, loop control structure, case control

structure, functions, subroutines, scope and lifetime of identifiers, parameter passing mechanism, arrays and strings,

Pointers, Pointers to Function, Function returning Pointers. (10 L)

Note: (Lab for remaining 1 Credits=30 Hrs.)


24

1. Concepts of Programming Language by Robert W. Sebesta, Addison Wesley, pearson Education Asia,

1999.

2. How to Program C by Deitel and Deitel, Addison Wesley, Pearson Education Asia.

3. Introduction to Computer Science by Ramon A. Mata-Toledo and Pauline K. Cushman, Mc Graw Hill

International Edition.

4. Programming Languages by D. Appleby and JJ Vande Kopple, Tata McGraw Hill, India.

5. C Programming a Modern Approach by KN King, WW Norton & Co.

6. Programming in C by Yashwant Kanetkar, B.P.B Publications.

26. SPECIAL FUNCTIONS LTP:3+1+0

UNIT-I: Beta and Gamma Functions, Euler Reflection Formula, Stirling‟s Asymptotic Formula, Gauss‟s

Multiplication Formula, Ratio of two gamma functions, Integral Representations for Logarithm of Gamma function

and Beta functions. (15L)

UNIT-II. Hypergeometric Differential Equations, Gauss Hypergeometric Function, Elementary Properties,

Conditions of convergence, Integral Representation, Gauss Theorem, Vandermonde‟s theorem, Kummer‟s theorem,

Linear transformation, Generalized Hypergeometric Functions, Elementary Properties, Integral Representation.

(15L)

UNIT-III: Legendre polynomials and functions, Solution of Legendre‟s differential equations, Generating

Functions, Rodrigue‟s Formula, Orthogonality of Legendre polynomials, Recurrence relations.

Bessel functions, Bessel differential equation and it‟s solution, Recurrence relation, Generating functions, Integral

representation. (15L)


1. G. E. Andrews, R. Askey, Ranjan Roy, Special Functions, Encyclopedia of Mathematics and its

Applications, Cambridge University Press, 1999.

2. E. D. Rainville, Special Functions, Macmillan, New York, 1960.

central university of rajasthan semester-wise scheme...

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